Geometry of the parallelism in polar spine spaces and their line reducts

Abstract

The concept of the spine geometry over a polar Grassmann space belongs to a wide family of partial affine line spaces. It is known that the geometry of a spine space over a projective Grassmann space can be developed in terms of points, so called affine lines, and their parallelism (in this case the parallelism is not intrinsically definable as it is not Veblenian). This paper aims to prove an analogous result for the polar spine spaces. As a by-product we obtain several other results on primitive notions for the geometry of polar spine spaces.